By moving from fixed rate stable models to changing rate: hyperstable models, the proposed work seeks to realize a quantum improvement in the ability of demographic models to capture observed behavior. Building on a recent discovery by the Principal Investigators, it will explore and develop the link between a birth sequence and the sequence of fertility schedules that preserve a fixed proportional distribution of births by age of mother. The work will involve basic research in mathematical demography, but will have applications to demographic modeling and estimation. It will proceed in 4 principal directions, specifically: l. Hyperstable Dynamics. The objective is to understand the mathematical foundations of hyperstable models and to explicate the major relationships between model functions. 2. Hyperstable Estimation. The objective is to improve and extend existing stable population modeling and estimation techniques through the introduction of models with changing vital rates. 3. Multistate Hyperstable Models. The objective is to generalize single state hyperstable models to the multistate case, and closely examine how they can reflect observed changes in interstate transfer rates. 4. Interacting Populations. Hyperstability can add a new dimension to models where demographic rates vary endogenously. We will construct and analyze "two-sex" hyperstable marriage and fertility models, and examine marriage squeezes in a hyperstable context. An interacting population/economy labor force model will be extended to the hyperstable case where the changing supply of and demand for labor can be examined. Ultimately, we seek to gain the fullest possible understanding of the interconnections between population structures and vital rates.